Question

The figure shows a system consisting of (i) a ring of outer radius 3R rolling clockwise without slipping on a horizontal surface with angular speed $?$ and (ii) an inner disc of radius 2R rotating anti-clockwise with angular speed $?/2$. The ring and disc are separated by frictionless ball bearings. The system is in the x-z plane. The point P on the inner disc is at a distance R from the origin, where OP makes an angle of 30$^{\circ}$ with the horizontal. Then with respect to the horizontal surface,

• A. the point O has a linear velocity $3R?\widehat {i}$
• B. the point P has a linear velocity $\frac{11}{4}R?\widehat {i}+\frac{\sqrt3}{4}R?\widehat {k}$
• C. the point P has a linear velocity $\frac{13}{4}R?\widehat {i}-\frac{\sqrt3}{4}R?\widehat {k}$
• D. the point P has a linear velocity $\bigg(3-\frac{\sqrt3}{4}\bigg)R?\widehat {i}+\frac{1}{4}R?\widehat {k}$

Answer 1

Answer: (B)

the point P has a linear velocity $\frac{11}{4}R?\widehat {i}+\frac{\sqrt3}{4}R?\widehat {k}$

Answer 2

Velocity of point O is $v_0=(3r?)\widehat {i}$ $v_{PO}$ is $\frac{R.?}{2}$ in the direction shown in figure. In vector form $V_{PO}=-\frac{R?}{4}sin 30^{\circ}\widehat {i}+\frac{R?}{2}cos 30^{\circ}\widehat {k}$ $=-\frac{R?}{4}\widehat {i}+\frac{\sqrt3R?}{4}\widehat {k}$ But $V_{PO}=v_P-v_O$ $\therefore V_P=v_{PO}-v_O$ $=\bigg(-\frac{R?}{4}\widehat {i}+\frac{\sqrt3R?}{4}\widehat {k}\bigg)+3R?\widehat {i}$ $=\frac{11}{4}R?\widehat {i}+\frac{\sqrt3}{4}R?\widehat {k}$